Example 3.6 Besides the situation considered in Theorem REF , there are other interesting classes of morphisms \(f:X \rightarrow Y\)  (and constructible functions \(\varphi \in F(X)\)  defined on their domain) satisfying the assumptions of Theorem REF . For instance, if \(\varphi \in F(X)\)  comes from a perverse sheaf, let \(f:X \rightarrow Y\)  be a morphism such that \(Rf_*\)  preserves perverse sheaves (e.g., a closed embedding or a quasi-finite affine morphism); then \(f_*(\varphi )\in F(Y)\)  also comes from a perverse sheaf and hence, since \(Y\)  is assumed smooth, \(CC(f_*(\varphi ))\)  is effective in \(T^*Y\) . Another example is provided by the projection \(f:X \rightarrow Y\)  from a semi-abelian variety \(X\)  onto its abelian part \(Y\) . In this case, it was shown in [1]} that if \(\varphi \in F(X)\)  comes from a perverse sheaf, then \(f_*(\varphi )\in F(Y)\)  also comes from a perverse sheaf. Hence \(CC(f_*(\varphi ))\)  is effective, and Theorem REF  applies to give \(\chi (X,\varphi ) \ge 0\) , an inequality initially proved in [2]}. For more examples in this direction, the interested reader may consult [1]}.

